ism5 - Chapter 5 Applications of Vectors in R2 and R3...

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Chapter 5 Applications of Vectors in R 2 and R 3 (Optional) Section 5.1, p. 263 2. (a) 4 i + j +4 k . (b) 3 i 8 j k . (c) 0 i +0 j k . (d) 4 i j +8 k . 10. 1 2 90. 12. 1. T.1. (a) Interchange of the second and third rows of the determinant in (2) changes the sign of the determinant. (b) ¯ ¯ ¯ ¯ ¯ ¯ ijk u 1 u 2 u 3 v 1 + w 1 v 2 + w 2 v 3 + w 3 ¯ ¯ ¯ ¯ ¯ ¯ = ¯ ¯ ¯ ¯ ¯ ¯ u 1 u 2 u 3 v 1 v 2 v 3 ¯ ¯ ¯ ¯ ¯ ¯ + ¯ ¯ ¯ ¯ ¯ ¯ u 1 u 2 u 3 w 1 w 2 w 3 ¯ ¯ ¯ ¯ ¯ ¯ . (c) Similar to proof for (b). (d) Follows from the homogeneity property for determinants: Theorem 3.5. (e) Follows from Theorem 3.3. (f) Follows from Theorem 3.4. (g) First let u = i and verify that the result holds. Similarly, let u = j and then u = k . Finally, let u = u 1 i + u 2 j + u 3 k . (h) First let w = i . Then ( u × v ) × i =( u 1 v 2 u 2 v 1 ) j ( u 3 v 1 u 1 v 3 ) k = u 1 ( v 1 i + v 2 j + v 3 k ) v 1 ( u 1 i + u 2 j + u 3 k ) = u 1 v v 1 u i · u ) v ( i · v ) u . Thus equality holds when w = i . Similarly it holds when w = j , when w = k , and (adding scalar multiples of the three equations), when w = w 1 i + w 2 j + w 3 k . T.2. ( u × v ) · w = ¯ ¯ ¯ ¯ ¯ ¯ w 1 w 2 w 3 u 1 u 2 u 3 v 1 v 2 v 3 ¯ ¯ ¯ ¯ ¯ ¯ = ¯ ¯ ¯ ¯ ¯ ¯ u 1 u 2 u 3 v 1 v 2 v 3 w 1 w 2 w 3 ¯ ¯ ¯ ¯ ¯ ¯ = u · ( v × w )
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80 Chapter 5 T.3. We have j × i = ¯ ¯ ¯ ¯ ¯ ¯ ijk 010 100 ¯ ¯ ¯ ¯ ¯ ¯ = k ¯ ¯ ¯ ¯ 01 10 ¯ ¯ ¯ ¯ = k k × j = ¯ ¯ ¯ ¯ ¯ ¯ 001 ¯ ¯ ¯ ¯ ¯ ¯ = i ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ = i i × k = ¯ ¯ ¯
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This note was uploaded on 06/25/2008 for the course MATH 22a taught by Professor Chuchel during the Spring '08 term at UC Davis.

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ism5 - Chapter 5 Applications of Vectors in R2 and R3...

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